Theorem bj-dvdemo2 33132

Description: Remove dependency on ax-13 2392 from dvdemo2 5053 (this removal is noteworthy since dvdemo1 5052 and dvdemo2 5053 illustrate the phenomenon of bundling). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-dvdemo2	⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)

Distinct variable group:   𝑥,𝑧

Proof of Theorem bj-dvdemo2

Step	Hyp	Ref	Expression
1		bj-el 33125	. 2 ⊢ ∃𝑥 𝑧 ∈ 𝑥
2		ax-1 6	. 2 ⊢ (𝑧 ∈ 𝑥 → (𝑥 = 𝑦 → 𝑧 ∈ 𝑥))
3	1, 2	eximii 1913	1 ⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)

Syntax hints:   → wi 4  ∃wex 1853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-pow 4993

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-nf 1859

This theorem is referenced by: (None)